Does the Riemann zeta function satisfy a differential equation?
Abbreviated Journal Title
J. Number Theory
Riemann zeta function; Infinite order differential equation; Euler-MacLauren summation formula; Mathematics
In Hilbert's 1900 address at the International Congress of Mathematicians, it was stated that the Riemann zeta function is the solution of no algebraic ordinary differential equation on its region of analyticity. It is natural, then, to inquire as to whether zeta(z) satisfies any non-algebraic differential equation. In the present paper, an elementary proof that zeta(z) formally satisfies an infinite order linear differential equation with analytic coefficients, T[zeta - 1] = 1/(z - 1), is given. We also show that this infinite order differential operator T may be inverted, and through inversion of T we obtain a series representation for zeta(z) which coincides exactly with the Euler-MacLauren summation formula for zeta(z). Relations to certain known results and specific values of zeta(z) are discussed. (C) 2014 Elsevier Inc. All rights reserved.
Journal of Number Theory
"Does the Riemann zeta function satisfy a differential equation?" (2015). Faculty Bibliography 2010s. 6843.